* You should build your own computer because it's fun, not that hard, gives a sense of satisfaction, and at the budget you're looking at, will save you many hundreds of dollars (or it will perform hundreds of dollars better, your choice).
* I'd recommend desktop over laptop - far and away better performance and high end gaming laptops are big, heavy, and produce a lot of heat. Not a bad idea to spend ~400-500 on a peppy little surfing laptop if a pad or phone doesn't fulfill your mobile computing needs.
* You want an Intel processor. AMD processors are good for many things, but not top-performance gaming. You also want a quad-core processor because games can only effectively utilize 4 cores. You probably want to overclock. Again, not that hard, very beneficial, and the only way to really get top performance. The exception to the quad-core rule is if you're rendering graphics, producing a stream, or that sort of thing. For just games, stick to quad-core and ignore hyperthreading (i7s= hyperthreading). The upshot? You want an i5-2500k or an i5-3570k. The i5-2500k is the current overclockable intel quad-core best for gaming (from the current-gen set of processors code named "sandy bridge." The i5-3570k will be released at the end of April, if you want to wait that long. It will be the overclockable intel quad-core best for gaming from the upcoming generation of intel processors code named "ivy brdige." The performance is expected to be essentially the same, with probably a 5-10% speed increase (less noticeable actual effect). The downside of the ivy bridge is you have to wait until the end of april.
* Your biggest decision with a budget of that size is actually what kind of monitor setup do you want to play on? There are three main luxury options:
1) A 23" or 24" 120hz monitor. Standard monitors these days can do only 60 frames per second. The new 120hz monitors can do up to 120 frames per second (and in some case, a funky form of 3D gaming at 60 frames per second that no one seems to like). Many proponents say that the faster frames make motion in game seem much smoother and love their 120hz monitors. Frankly, I personally don't see much of a difference. Cost of decent 120hz monitor: $450-550 or so. Example: BenQ XL2420.
2) A BIG monitor. At 27" or 30", these monitors still look sharp because of an increased resolution (2560x1440 or 2560x1600). Just like a normal quality monitor. Only really freaking big. This is probably the option I'd go for. Cost can range from around $400 for the cheaper 27" imports to $1200 or so for the better 30" monitors. Example: HP zr30w.
3) THREE monitors. Ordinary monitors can be placed side-by-side for multi-monitor gaming (often referred to as Eyefinity or Surround, two brand names for video card drivers). This gives you a very wide screen @ 5760x1080 resolution. Cost depends on the quality of the monitor. ~$350-700. Example: 3x Dell U2312HM.
4) Just keep a single normal monitor like whatever you have right now. In that case, you have too much budget - you can achieve maximum performance on a single monitor for ~$1100 or so in computer tower, WAY under your budget. If you're not a mega gamer, not a bad idea. Example: 1x Dell U2312HM.
* This website being hardware-oriented, we tend to ignore peripherals and assume you already have the monitor setup you want. But it may actually be your most important decision. Certainly, the super-quality graphics cards everyone will recommend when parts lists start flying will be total overkill unless you have one of the first three options above.
* Budget for other peripherals. A luxury keyboard, mouse, headphones, maybe some speakers, can cost up to $300 (all together) or even more.
For a kick-ass tower capable of smoothly running one of the 3 luxury monitor setups above, you'll be looking at something like this:
(~$400-$450) intel i5-XXXXk with either a z68 (sandy bridge) or z77 (ivy bridge) + aftermarket CPU cooler.
(~$350) case, psu, & RAM - a luxury case, of which there are many (I recommend the corsair 550d), will run about $150, the PSU will be about as much (or more if you budget for 3 video cards), and ram should be about $50
(~$320) SSD, HDD, and DVD-burner. A DVD burner is dirt cheap. A 128GB SSD, a storage drive that reduces boot time, load times, and general access wait time, will cost about $170. More data storage than you really need (2TB or so) will cost about $120 in the form of a hard drive.
(~$1000) Two top-end video cards (7970s or 680s). One will manage the luxury monitor solutions above fairly well, 2 will make them shine. 3 is a bit out of your budget, not really necessary, and a pain to keep cool. If you are NOT getting one of the luxury monitor setups above, one top-end video card will be more than enough for your purposes.
The above estimates are basicaly high-end & luxury everything for a gaming setup. Now you may be adding up numbers and saying, hey, those estimates and a 30" monitor come to about $3200. Frankly, you can save money for about the same performance in many places in the build (and waste it... which will often come up in the advice you get).
Hopefully that overview can give you an idea of how to start looking at what you're purchasing.
Send a PM if you need tech talk translated to something more understandable or want me to get more specific about what you're looking at.